You're right. Saying "hey Cortana" takes twice as long as saying "Xbox". This may not matter if you're doing something like inviting someone to a party, or launching a game, but when you're trying to pause video or take a screenshot, it's really noticeable. Cortana is also apparently taking longer for most commands than she should, but I have my doubts about how fast she'll be once it's fixed. I've too been wondering if these commands are being send out across the internet. If this is being done with basic system commands, then... wow.
Old setup:
"Xbox, take a screenshot" and the popup displays less than a second after I finish the last word.
Current setup:
"Hey Cortana, take a screenshot" and after a second or two you see her going through each word you just said, one at a time. After 5 seconds (I timed it, multiple times), the popup confirming the screenshot shows up. This is of course after a she gives me a Bing search for "Taken" a couple times.
The biggest issue I have with all of this is that I don't understand what the net gain is. Even if they get Cortana to work as fast as the old commands, you're still using a longer phrase for the same things. There were already commands in for a lot of the common things you might want Cortana for. If there was a need for more commands, why not just add more commands? Of course, I know what the answer probably is. People like the voice commands, but won't buy a Kinect. Cortana allows them to use their microphone. I feel like a built in microphone in the Xbox One S and Scorpio would have been a better solution though.